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Place a rack in the lower third of your oven, then preheat the oven to 400 degrees F. Line a large, rimmed baking sheet with foil and coat the foil with nonstick spray.
Place the carrots and potatoes in the center of the baking sheet. Drizzle with the olive oil and honey, then sprinkle with cumin, smoked paprika, salt, and cayenne. Toss to coat, then spread into a single layer.
Bake in the oven’s lower third until vegetables are browned in spots and tender when pierced with a small, sharp knife, about 25 to 30 minutes, tossing halfway through. Sprinkle with rosemary and toss to coat.
While the vegetables bake, prepare the honey yogurt sauce: In a small bowl, stir together the Greek yogurt and honey. Transfer the vegetables to a serving plate. Serve warm with big dollops of yogurt on top and extra yogurt on the side.